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Cambridge University Press 978-1-107-05764-7 — Structural Geology Haakon Fossen Frontmatter More Information www.cambridge.org © in this web service Cambridge University Press Structural Geology Second Edition Tis market-leading textbook has been fully updated in response to extensive user feed- back. It includes a new chapter on joints and veins, additional examples from around the world, stunning new feld photos, and extended online resources with new animations and exercises. Te book’s practical emphasis, hugely popular in the frst edition, features applications in the upper crust, including petroleum and groundwater geology, highlight- ing the importance of structural geology in exploration and exploitation of petroleum and water resources. Carefully designed full-color illustrations work closely with the text to support student learning, and are supplemented with high-quality photos from around the world. Examples and parallels drawn from practical everyday situations engage students, and end-of-chapter review questions help them to check their understanding. Updated e-learning modules are available online for most chapters and further reinforce key topics using summaries, innovative animations to bring concepts to life, and addi- tional examples and fgures. Haakon Fossen is Professor of Structural Geology at the University of Bergen, Norway, where he is afliated with the Department of Earth Science and the Natural History Collections. His professional career has involved work as an exploration and production geologist/geophysicist for Statoil and as a Professor at the University of Bergen (1996 to present), in addition to periods of geologic mapping and mineral exploration in Norway. His research ranges from hard to sof rocks and includes studies of folds, shear zones, formation and collapse of the Caledonian Orogen, numerical modeling of deformation (transpression), the evolution of the North Sea rif, and studies of deformed sandstones in the western United States. He has conducted extensive feld work in various parts of the world, notably Norway, Utah/Colorado, and Sinai, and his research is based on feld mapping, microscopy, physical and numerical modeling, geochronology and seismic in- terpretation. Professor Fossen has been involved in editing several international geology journals, has authored over 100 scientifc publications, and has written two other books and several book chapters. He has taught undergraduate structural geology courses for twenty years and has a keen interest in developing electronic teaching resources to aid student visualization and understanding of geological structures.
Transcript
Page 1: Structural Geology - Cambridge University Press...Cambridge University Press 978-1-107-05764-7 — Structural Geology Haakon Fossen Frontmatter More Information © in this web service

Cambridge University Press978-1-107-05764-7 — Structural GeologyHaakon Fossen FrontmatterMore Information

www.cambridge.org© in this web service Cambridge University Press

Structural GeologySecond Edition

This market-leading textbook has been fully updated in response to extensive user feed-back. It includes a new chapter on joints and veins, additional examples from around the world, stunning new field photos, and extended online resources with new animations and exercises. The book’s practical emphasis, hugely popular in the first edition, features applications in the upper crust, including petroleum and groundwater geology, highlight-ing the importance of structural geology in exploration and exploitation of petroleum and water resources. Carefully designed full-color illustrations work closely with the text to support student learning, and are supplemented with high-quality photos from around the world. Examples and parallels drawn from practical everyday situations engage students, and end-of-chapter review questions help them to check their understanding. Updated e-learning modules are available online for most chapters and further reinforce key topics using summaries, innovative animations to bring concepts to life, and addi-tional examples and figures.

Haakon Fossen is Professor of Structural Geology at the University of Bergen, Norway, where he is affiliated with the Department of Earth Science and the Natural History Collections. His professional career has involved work as an exploration and production geologist/geophysicist for Statoil and as a Professor at the University of Bergen (1996 to present), in addition to periods of geologic mapping and mineral exploration in Norway. His research ranges from hard to soft rocks and includes studies of folds, shear zones, formation and collapse of the Caledonian Orogen, numerical modeling of deformation (transpression), the evolution of the North Sea rift, and studies of deformed sandstones in the western United States. He has conducted extensive field work in various parts of the world, notably Norway, Utah/Colorado, and Sinai, and his research is based on field mapping, microscopy, physical and numerical modeling, geochronology and seismic in-terpretation. Professor Fossen has been involved in editing several international geology journals, has authored over 100 scientific publications, and has written two other books and several book chapters. He has taught undergraduate structural geology courses for twenty years and has a keen interest in developing electronic teaching resources to aid student visualization and understanding of geological structures.

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Structural Geology

SECOND EDITION

Haakon FossenUNIVERSITY OF BERGEN, NORWAY

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University Printing House, Cambridge CB2 8BS, United Kingdom

Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.

www.cambridge.org Information on this title: www.cambridge.org/9781107057647

© Haakon Fossen 2016

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2016

Printed in the United Kingdom by Clays, St Ives plc

A catalogue record for this publication is available from the British Library

ISBN 978-1-107-05764-7 Hardback

Additional resources for this publication at www.cambridge.org/fossen2e

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Reprinted 2016

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How to use this book viii

Preface xi

Acknowledgments xii

List of symbols xiii

1 Structural geology and structural analysis 11.1 Approaching structural geology 21.2 Structural geology and tectonics 21.3 Structural data sets 41.4 Field data 51.5 Remote sensing and geodesy 81.6 DEM, GIS and Google Earth 101.7 Seismic data 101.8 Experimental data 141.9 Numerical modeling 151.10 Other data sources 151.11 Organizing the data 161.12 Structural analysis 181.13 Concluding remarks 22

2 Deformation 252.1 What is deformation? 262.2 Components of deformation 272.3 System of reference 282.4 Deformation: detached from history 292.5 Homogeneous and heterogeneous

deformation 292.6 Mathematical description of deformation 302.7 One-dimensional strain 302.8 Strain in two dimensions 322.9 Three-dimensional strain 332.10 The strain ellipsoid 342.11 More about the strain ellipsoid 352.12 Volume change 362.13 Uniaxial strain (compaction) 372.14 Pure shear and coaxial deformations 382.15 Simple shear 382.16 Subsimple shear 392.17 Progressive deformation and flow

parameters 392.18 Velocity field 41

2.19 Flow apophyses 422.20 Vorticity and Wk 432.21 Steady-state deformation 452.22 Incremental deformation 452.23 Strain compatibility and boundary conditions 452.24 Deformation history from

deformed rocks 462.25 Coaxiality and progressive simple shear 472.26 Progressive pure shear 492.27 Progressive subsimple shear 502.28 Simple and pure shear and their scale

dependence 512.29 General three-dimensional deformation 512.30 Stress versus strain 52 Summary 55

3 Strain in rocks 593.1 Why perform strain analysis? 603.2 Strain in one dimension 603.3 Strain in two dimensions 603.4 Strain in three dimensions 67 Summary 70

4 Stress 734.1 Definitions, magnitudes and units 744.2 Stress on a surface 744.3 Stress at a point 754.4 Stress components 774.5 The stress tensor (matrix) 774.6 Deviatoric stress and mean stress 784.7 Mohr circle and diagram 79 Summary 80

5 Stress in the lithosphere 835.1 Importance of stress measurements 845.2 Stress measurements 845.3 Reference states of stress 875.4 The thermal effect on horizontal stress 915.5 Residual stress 925.6 Tectonic stress 92

Contents

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ContentsVI

9.5 The birth and growth of faults 1939.6 Growth of fault populations 2049.7 Faults, communication and sealing properties 210 Summary 216

10 Kinematics and paleostress in the brittle regime 22110.1 Kinematic criteria 22210.2 Stress from faults 22410.3 A kinematic approach to fault slip data 22710.4 Contractional and extensional structures 230 Summary 231

11 Deformation at the microscale 23511.1 Deformation mechanisms and

microstructures 23611.2 Brittle versus plastic deformation

mechanisms 23611.3 Brittle deformation mechanisms 23711.4 Mechanical twinning 23711.5 Crystal defects 23911.6 From the atomic scale to

microstructures 245 Summary 254

12 Folds and folding 25712.1 Geometric description 25812.2 Folding: mechanisms and processes 26512.3 Fold interference patterns and refolded folds 27412.4 Folds in shear zones 27612.5 Folding at shallow crustal depths 277 Summary 278

13 Foliation and cleavage 28313.1 Basic concepts 28413.2 Relative age terminology 28613.3 Cleavage development 28613.4 Cleavage, folds and strain 29113.5 Foliations in quartzites, gneisses and

mylonite zones 295 Summary 297

14 Lineations 30114.1 Basic terminology 30214.2 Lineations related to plastic deformation 302

5.7 Global stress patterns 945.8 Differential stress, deviatoric stress and

some implications 97 Summary 98

6 Rheology 1016.1 Rheology and continuum mechanics 1026.2 Idealized conditions 1026.3 Elastic materials 1036.4 Plasticity and flow: permanent deformation 1076.5 Combined models 1116.6 Experiments 1136.7 The role of temperature, water, etc. 1146.8 Definition of plastic, ductile and brittle

deformation 1166.9 Rheology of the lithosphere 117 Summary 119

7 Fracture and brittle deformation 1237.1 Brittle deformation mechanisms 1247.2 Types of fractures 1257.3 Failure and fracture criteria 1297.4 Microdefects and failure 1347.5 Fracture termination and interaction 1387.6 Reactivation and frictional sliding 1407.7 Fluid pressure, effective stress and

poroelasticity 1417.8 Deformation bands and fractures in

porous rocks 143 Summary 149

8 Joints and veins 1538.1 Definition and characteristics 1548.2 Kinematics and stress 1568.3 How, why and where joints form 1578.4 Joint distributions 1618.5 Growth and morphology of joints 1648.6 Joint interaction and relative age 1668.7 Joints, permeability and fluid flow 1678.8 Veins 168 Summary 174

9 Faults 1779.1 Fault terminology 1789.2 Fault anatomy 1839.3 Displacement distribution 1879.4 Identifying faults in an oil field setting 188

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Contents VII

19 Strike-slip, transpression and transtension 40119.1 Strike-slip faults 40219.2 Transfer faults 40219.3 Transcurrent faults 40419.4 Development and anatomy of strike-slip

faults 40519.5 Transpression and transtension 41019.6 Strain partitioning 413 Summary 414

20 Salt tectonics 41720.1 Salt tectonics and halokinesis 41820.2 Salt properties and rheology 41820.3 Salt diapirism, salt geometry and the

flow of salt 42020.4 Rising diapirs: processes 42920.5 Salt diapirism in the extensional regime 43020.6 Diapirism in the contractional regime 43220.7 Diapirism in strike-slip settings 43520.8 Salt collapse by karstification 43520.9 Salt décollements 436 Summary 438

21 Balancing and restoration 44121.1 Basic concepts and definitions 44221.2 Restoration of geologic sections 44221.3 Restoration in map view 44721.4 Geomechanically based restoration 45021.5 Restoration in three dimensions 45121.6 Backstripping 451 Summary 452

22 A glimpse of a larger picture 45522.1 Synthesizing 45622.2 Deformation phases 45622.3 Progressive deformation 45722.4 Metamorphic textures 45722.5 Radiometric dating and P–T–t paths 46022.6 Tectonics and sedimentation 461 Summary 462

Appendix A: More about the deformation matrix 464

Appendix B: Spherical projections 468

Glossary 474

References 495

Cover and chapter image captions 501

Index 503

14.3 Lineations in the brittle regime 30614.4 Lineations and kinematics 308 Summary 311

15 Boudinage 31515.1 Boudinage and pinch-and-swell

structures 31615.2 Geometry, viscosity and strain 31615.3 Asymmetric boudinage and rotation 31915.4 Foliation boudinage 32015.5 Boudinage and the strain ellipse 32215.6 Large-scale boudinage 323 Summary 325

16 Shear zones and mylonites 32916.1 What is a shear zone? 33016.2 The ideal plastic shear zone 33316.3 Adding pure shear to a simple

shear zone 33716.4 Non-plane strain shear zones 34016.5 Mylonites and kinematic indicators 34116.6 Growth of shear zones 349 Summary 351

17 Contractional regimes 35517.1 Contractional faults 35617.2 Thrust faults 35717.3 Ramps, thrusts and folds 36217.4 Orogenic wedges 368 Summary 373

18 Extensional regimes 37718.1 Extensional faults 37818.2 Fault systems 37918.3 Low-angle faults and core complexes 38118.4 Ramp-flat-ramp geometries 38618.5 Footwall versus hanging-wall collapse 38718.6 Rifting 38818.7 Half-grabens and accommodation

zones 38918.8 Pure and simple shear models 38918.9 Stretching estimates, fractals and

power-law relations 39018.10 Passive margins and oceanic rifts 39218.11 Orogenic extension and orogenic

collapse 39318.12 Postorogenic extension 395 Summary 396

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Contractional faults occur in any tectonic regime,

but they are most common along destructive plate

boundaries and in intracratonic orogenic zones.

Contractional structures received much attention

from the last part of the nineteenth century up to

the end of the twentieth century, when the focus

shifted somewhat towards extensional structures.

The study of contractional faults resulted in the de-

velopment of balanced cross-sections, and brought

attention to the role of fault overlaps and relay

structures, the relation between displacement and

fault length, and the mechanical aspects of fault-

ing. Understanding contractional faults is important

not only for better understanding of orogenic pro-

cesses in general, but also for improved petroleum

exploration methods, because a number of the

world’s oil resources are located in fold and thrust

belts. The fundamentals of contractional faults and

related structures are covered in this chapter, with a

focus on thrust structures found in orogenic belts.

17Contractional regimes

The e-module for this chapter, Contraction,

provides further support on the following

topics:

Each chapter starts with a general introduction,

which presents a context for the topic within

structural geology as a whole. These introductions

provide a roadmap for the chapter and will help

you to navigate through the book. The box alongside

identifies which online e-module accompanies the

chapter and the topics that it covers.

How to use this book

The main text contains highlighted terms and key

expressions that you will need to understand and become

familiar with. Many of these terms are listed in the

Glossary at the back of the book. The Glossary allows you

to easily look up terms whenever needed and can also be

used to review important topics and key facts. Each chapter

also contains a series of highlighted statements to

encourage you to pause and review your understanding of

what you have read.

Boxes present in-depth information about a particular

subject, helpful examples or relevant background informa-

tion. Other important points are brought together in the

chapter summaries. Review questions should be used

to test your understanding of the chapter before moving

on to the next topic. Answers to these questions are given

on the book’s webpage. Further reading sections provide

references to selected papers and books for those interested

in more detail or advanced information.

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How to use this book IX

Online resourceswww.cambridge.org/fossen2e

Specially prepared resources, unique to this book, are

available from the book’s webpage. These include:

• E-learning modules that combine animations, text,

illustrations and photographs. These present key

aspects of structural geology in a highly visual and

interactive environment.

• Answers to the end-of-chapter review questions

for instructors.

• Additional student exercises (with solutions for

instructors).

• All of the figures for each chapter as jpeg and

PowerPoint files.

• An electronic glossary of terms.

• A gallery of supplementary figures illustrating

additional geologic structures and field examples.

• Tutorial videos from the field.

• Links to other web-based structural geology

resources including software.

• Links to the author’s blog and community Face-

book page.

E-learning modules further reinforce key topics

using summaries, additional examples and figures,

and innovative animations to bring concepts to life.

Use of these e-modules is highly recommended after

reading the chapter as part of review and exam

preparation. The modules provide supplementary

information that complements the main text.

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Preface

structural geology text of the twenty-first century should come with specially prepared e-learning resources, so the package of e-learning material that is presented with this book should be regarded as part of the present book concept.

Book structureThe structure of the book is in many ways traditional, go-ing from strain (Chapters 2 and 3) to stress (Chapters 4 and 5) and via rheology (Chapter 6) to brittle deforma-tion (Chapters 7–10). Of these, Chapter 2 contains some mater ial that would be too detailed and advanced for some students and classes, but selective reading is possible. Then, after a short introduction to the microscale struc-tures and processes that distinguish crystal-plastic from brittle deformation (Chapter 11), ductile deformation structures such as folding, boudinage, foliations and shear zones are discussed (Chapters 12–16). Three consecutive chapters then follow that are founded on the three prin-cipal tectonic regimes (Chapters 17–19) before salt tec-tonics and restoration principles are presented (Chapters 20 and 21). A final chapter, where links to metamorphic petrology as well as stratigraphy are drawn, rounds off the book, and suggests that structural geology and tectonics largely rely on other disciplines. The chapters do not have to be read in numerical order, and most chapters can be used individually.

Emphasis and examplesThe book seeks to cover a wide ground within the field of structural geology, and examples presented in the text are from different parts of the world. However, pictures and illustrations from a few geographic areas reappear. One of those is the North Sea rift system, which I know from my years with the Norwegian oil company Statoil and later academic research. Another is the Colorado Plateau, which over the last two decades has become one of my favorite places to do field work. A third, and much wetter and greener one, is the Scandinavian Caledonides, bal-anced by the much hotter Araçuai-Ribeira Belt in Brazil. Many of the examples used to illustrate structures typical of the plastic regime come from these orogenic belts.

This is the second edition of Structural Geology; a textbook that was first published in 2010. The first edition was very well received among students, lecturers and industry pro-fessionals alike. I received a lot of encouraging comments and helpful feedback from readers, and this has been a motivating factor for preparing a new and improved ver-sion with updated text, illustrations and photographs that preserves the overall structure of the previous edition.

The purpose of the book is to introduce undergraduate students, and others with a general geologic background, to the basic principles, aspects and methods of structural geology. It is mainly concerned with the structural geol-ogy of the crust, although the processes and structures described are relevant also for deformation that occurs at deeper levels within our planet. Further, remote data from Mars and other planets indicate that many aspects of terrestrial structural geology are relevant also beyond our own planet.

The field of structural geology is very broad, and the content of this book presents a selection of important subjects within this field. Making the selection has not been easy, knowing that lecturers tend to prefer their own favorite aspects of, and approaches to, structural geology, or make selections according to their local departmental course curriculum. Existing textbooks in structural geology tend to emphasize the ductile or plastic deformation that occurs in the middle and low-er crust. In this book I have tried to treat the frictional regime in the upper crust more extensively so that it bet-ter balances that of the deeper parts of the crust, which makes some chapters particularly relevant to courses where petroleum geology and brittle deformation in general are emphasized. This philosophy is extended with the second edition, particularly by the addition of a new chapter on joints and veins.

Obtaining this balance was one of several motivating factors for writing this book, and is perhaps related to my mixed petroleum geology and hard-rock structural geol-ogy experience. Other motivating factors include the de-sire to make a book where I could draw or redraw all of the illustrations and be able to present the first full-color book in structural geology. I also thought that a fundamental

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Acknowledgments

Kirkpatrick, Stephen Lippard, Christophe Pascal, Atle Rotevatn, Zoe Shipton, Holger Stunitz, Bruce Trudgill, Carolina Cavalcante, Luiz Morales and Fred Vollmer for critically reading various parts of the text. Valuable assist ance and company in the field that have influenced this book were provided by Julio Almeida, Renato Almei-da, Nicolas Badertscher, Wallace Bothner, Jean M. Cres-pi, Rui Dias, Marcos Egydio da Silva, Jim Evans, Jonny Hesthammer, Fernando O. Marques, Roger Soliva, John Walsh and Adolph Yonkee. Thanks also to readers who sent comments on various parts of the first editions. Fi-nally, other textbooks have been of invaluable worth both during my time as a student of structural geology in gen-eral and during the preparation of this book. In particu-lar I have enjoyed and learned a lot from the books by Hobbs, Means and Williams (1976), Twiss and Moores (2007), van der Pluijm and Marshak (2004), and various editions of George H. Davis and co-authors’ Structural

Geology text, as well as the excellent Microtectonics text by Passchier and Trouw (2005).

During the writing of this textbook I have built on ex-perience and knowledge achieved as a student, during various industrial and academic positions, and through the writing of this book. In this respect I want to thank fellow students, geologists and professors with whom I have interacted during my time at the Universities of Bergen, Oslo, Minnesota and Utah, at Utah State University, in Statoil and at the Geological Survey of Norway. In particular, my advisers and friends Tim Holst, Peter Hudleston and Christian Teyssier deserve special thanks for generously sharing their knowledge during my time as a student, and also once fellow student Basil Tikoff for valuable discussions and exchange of ideas in Pillsbury Hall. Among my many co-workers, colleagues and former students I wish to extend special thanks to Roy Gabrielsen, Jan Inge Faleide, Jonny Hesthammer, Rich Schultz, Roger Soliva, Gregory Ballas, Rob Gaw-thorpe, Ritske Huismans and Carolina Cavalcante.

Special thanks also go to Wallace Bothner, Rob But-ler, Nestor Cardozo, Declan DePaor, Jim Evans, James

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Symbols

a long axis of ellipse representing a microcrackA area;

empirically determined constant in flow lawsB layer thicknessc short axis of ellipse representing a microcrackC cohesion or cohesional strength of a rockCf cohesive strength of a faultd offsetdcl thickness of clay layerD displacement;

fractal dimensionDmax maximum displacement along a fault trace or on a fault surfaceD deformation (gradient) matrixe = ε elongatione=ε elongation rate (de / dt)ėx , ėy elongation rates in the x and y directions (s−1)e1, e2 , e3 eigenvectors of deformation matrix, identical to the three axes of the strain

ellipsoidē logarithmic (natural) elongationēs natural octahedral unit shearE Young’s modulus; activation energy for migration of vacancies through a crystal

(J mol−1 K−1)E* activation energyF force vector (kg m s−2, N)Fn normal component of the force vectorFs shear component of the force vectorg acceleration due to gravity (m/s2)h layer thicknessh0 initial layer thicknesshT layer thickness at onset of folding (buckling)ISA1−3 instantaneous stretching axesk parameter describing the shape of the strain ellipsoid

(lines in the Flinn diagram)K bulk modulusKi stress intensity factorKc fracture toughnesskx , ky pure shear components, diagonal elements in the pure shear and simple

shear matricesl line length (m)l0 line length prior to deformation (m)L velocity tensor (matrix)L fault length;

wavelength

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XIV List of symbols

Ld dominant wavelengthLT actual length of a folded layer over the distance of one wavelengthn exponent of displacement-length scaling lawpf fluid pressureP pressure (Pa)Q activation energyR ellipticity or aspect ratio of ellipse (long over short axis); gas constant

(J kg−1 K−1)Rf final ellipticity of an object that was non-circular prior to deformationRi initial ellipticity of an object (prior to deformation)Rs same as R, used in connection with the Rf/ϕ-method to distinguish

it from Rf

Rxy X/YRyz Y/Zs stretchingS stretching tensor, symmetric part of Lt time (s)T temperature (K or°C);

uniaxial tensile strength (bar); local displacement or throw of a fault when calculating SGR and SSF

v velocity vector (m/s)V volume (m3)V0 volume prior to deformationVp velocity of P-wavesVs velocity of S-wavesw vorticity vectorw vorticityW vorticity (or spin) tensor, which is the skew-symmetric component of LWk kinematic vorticity numberx vector or point in a coordinate system prior to deformationx' vector or point in a coordinate system after deformationx, y, z coordinate axes, z being verticalX, Y, Z principal strain axes; X ≥Y ≥ ZZ crustal depth (m)α thermal expansion factor (K−1);

Biot poroelastic parameter; angle between passive marker and shear direction at onset of non-coaxial deformation (Chapter 15); angle between flow apophyses (Chapter 2)

α' angle between passive marker and shear direction after a non-coaxial deformation

β stretching factor, equal to s∆ volume change factor∆σ change in stressγ shear strainγoct octahedral shear strainγ shear strain rateΓ non-diagonal entry in deformation matrix for subsimple shearη viscosity constant (N s m−2)λ quadratic elongationλ 1, λ2, λ3 eigenvalues of deformation matrix√λ1, √λ2, √λ3 length of strain ellipse axes

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XVList of symbols

μ shear modulus; viscosity

μf coefficient of sliding frictionμL viscosity of buckling competent layerμM viscosity of matrix to buckling competent layerν Poisson’s ratio;

Lode’s parameterθ angle between the normal to a fracture and σ1;

angle between ISA1 and the shear planeθ' angle between X and the shear planeρ density (g/cm3)σ stress (∆F/∆A) (bar: 1 bar = 1.0197 kg/cm2 = 105 Pa = 106 dyne/cm2)σ stress vector (traction vector)σ1 > σ2 > σ3 principal stressesσ effective stressσa axial stressσdev deviatoric stressσdiff differential stress (σ1 − σ3)σH max horizontal stressσh min horizontal stressσh* average horizontal stress in thinned part of the lithosphere

(constant-horizontal-stress model)σm mean stress (σ1 + σ2 + σ3)/3σn normal stressσr remote stressσs shear stressσt tectonic stressσtip stress at tip of fracture or point of max curvature along pore marginσtot total stress (σm + σdev)σv vertical stressσg

n normal stress at grain–grain or grain–wall contact areas in porous mediumσn

w average normal stress exerted on wall by grains in porous mediumϕ internal friction (rock mechanics);

angle between X and a reference line at onset of deformation (Rf/ϕ-method)ϕ' angle between X and a reference line after a deformation (Rf/ϕ-method)Φ porosityψ angular shearω angular velocity vector

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